Chapter XX of Treatise on Probability

Pure Induction

Source: A Treatise on Probability (1920) publ. MacMillan, 1948.
Just a short excerpt reproduced here.

... Pure Induction can be usefully employed to strengthen an argument
if, after a certain number of instances have been examined, we
have, from some other source, a finite probability in favour of
the generalisation, and, assuming the generalisation is false,
a finite uncertainty as to its conclusion being satisfied by the
next hitherto unexamined instance which satisfies its premise.
To take an example, Pure Induction can be used to support the
generalisation that the sun will rise every morning for the next
million years, provided that with the experience we have actually
had there are finite probabilities, however small, derived
from some other source, first, in favour of the generalisation,
and, second, in favour of the sun's not rising to-morrow assuming
the generalisation to be false. Given these finite probabilities,
obtained otherwise, however small, then the probability can be
strengthened and can tend to increase towards certainty by the
mere multiplication of instances provided that these instances
are so far distinct that they are not inferable one from another.

7.

Those supposed proofs of the Inductive Principle, which are
based openly or implicitly on an argument in inverse probability,
are all vitiated by unjustifiable assumptions relating to the
magnitude of the a priori probability p0. Jevons,
for instance, avowedly assumes that we may, in the absence of
-special information, suppose any unexamined hypothesis to be
as likely as not. It is difficult to see how such a belief, if
even its most immediate implications had been properly apprehended,
could have remained plausible to a mind of so sound a practical
judgment as his. The arguments against it and the contradictions
to which it leads have been dealt with in Chapter IV. The demonstration
of Laplace, which depends upon the Rule of Succession, will be
discussed in Chapter XXX.

8.

The prior probability, which must always be found, before the
method of pure induction can be usefully employed to support a
substantial argument, is derived, I think, in most ordinary cases
- with what justification it remains to discuss - from considerations
of Analogy. But the conditions of valid induction as they have
been enunciated above, are quite independent of analogy, and might
be applicable to other types of argument. In certain cases we
might feel justified in assuming directly that the necessary
conditions are satisfied.

Our belief, for instance, in the validity of a logical scheme
is based partly upon inductive grounds - on the number
of conclusions, each seemingly true on its own account, which
can be derived from the axioms - and partly on a degree of self-evidence
in the axioms themselves sufficient to give them the initial probability
upon which induction can build. We depend upon the initial presumption
that, if a proposition appears to us to be true, this is by itself,
in the absence of opposing evidence, some reason for its
being as well as appearing true. We cannot deny that what
appears true is sometimes false, but, unless we can assume some
substantial relation of probability between the appearance and
the reality of truth, the possibility of even probable knowledge
is at an end.

The conception of our having some reason, though not a
conclusive one, for certain beliefs, arising out of direct inspection,
may prove important to the theory of epistemology. The old metaphysics
has been greatly hindered by reason of its having always demanded
demonstrative certainty. Much of the cogency of Hume's criticism
arises out of the assumption of methods of certainty on the part
of those systems against which it was directed. The earlier realists
were hampered by their not perceiving that lesser claims in the
beginning might yield them what they wanted in the end. And transcendental
philosophy has partly arisen, I believe, through the belief that
there is no knowledge on these matters short of certain knowledge,
being combined with the belief that such certain knowledge of
metaphysical questions is beyond the power of ordinary methods.

When we allow that probable knowledge is, nevertheless, real,
a new method of argument can be introduced into metaphysical discussions.
The demonstrative method can be laid on one side, and we may
attempt to advance the argument by taking account of circumstances
which seem to give some reason for preferring one alternative
to another. Great progress may follow if the nature and reality
of objects of perception, for instance, can be usefully investigated
by methods not altogether dissimilar from those employed in science
and with the prospect of obtaining as high a degree of certainty
as that which belongs to some scientific conclusions; and it may
conceivably be shown that a belief in the conclusions of science,
enunciated in any reasonable manner however restricted, involves
a preference for some metaphysical conclusions over others.

9.

Apart from analysis, careful reflection would hardly lead us
to expect that a conclusion which is based on no other than grounds
of pure induction, defined as I have defined them as consisting
of repetition of instances merely, could attain in this way to
a high degree of probability. To this extent we ought all of
us to agree with Hume. We have found that the suggestions of
common sense are supported by more precise methods. Moreover,
we constantly distinguish between arguments, which we call inductive,
upon other grounds than the number of instances upon which they
are based; and under certain conditions we regard as crucial an
insignificant number of experiments. The method of pure induction
may be a useful means of strengthening a probability based on
some other ground. In the case, however, of most scientific arguments,
which would commonly be called inductive, the probability that
we are right, when we make predictions on the basis of past experience,
depends not so much on the number of past experiences upon which
we rely, as on the degree in which the circumstances of these
experiences resemble the known circumstances in which the prediction
is to take effect. Scientific method, indeed, is mainly devoted
to discovering means of so heightening the known analogy that
we may dispense as far as possible with the methods of pure induction.

When, therefore, our previous knowledge is considerable and the
analogy is good, the purely inductive part of the argument may
take a very subsidiary place. But when our knowledge of the instances
is slight, we may have to depend upon pure induction a good deal.
In an advanced science it is a last resort, the least satisfactory
of the methods. But sometimes it must be our first resort, the
method upon which we must depend in the dawn of knowledge and
in fundamental inquiries where we must presuppose nothing.